integrate-each-of-the-given-functions-int-frac12-d-x64x2-3

Question

Integrate each of the given functions.
$$\int \frac{12 d x}{64+x^{2}}$$

Tyler Moulton · Accepted Answer

we want to integrate the following expression, the integral of 12 X divided by 64 plus X square. This question is challenging our knowledge and ability to utilize integration techniques that we picked up throughout our journey. And single variable calculus in particular is testing our newfound knowledge of. The integral is known as inverse trig and metric integral. There are two inverse trig integral that are relevant. That I listen here the first and the last state of the integral. D&#x27;you over a squared minus U squared arsenio over a plus. See the second on the rights of the integral. D&#x27;you over A squared plus B squared is one over a R candidate over a plus C. We see that are integral is in the form to best if we identify you, do you end A. We can easily solve. So for a reason. Year old two we see that U. Is equal to X. Do you equal D. S. A. Is equal to route 64 or eight. Thus we can plug into our solution for equation to which means are integral is 12 integral dX 6/64 plus X squared, or 12/8 are tangent, X ray plus C. Or three. House are tangent X over eight plus C.

Amy Jiang · Answer

Okay, So notice here that inside our radical we haven&#x27;t x squared. And outside of that, we have a next. So we know that the derivative of X word is in terms of X. So we&#x27;ll start by letting you equal to X squared plus 12 x and now are derivative. That&#x27;s people to two X plus 12 the ex and all its factor out of two. So you get exports six and in D X, which is exactly what we want. So we&#x27;ll divide both sides by two. So we get to you over to Executive Fallen. Okay, I know. Let&#x27;s replace what we have. So we have the radical, which is actually a to the power of what happens. That&#x27;s you could part 1/2 and indeed over to taking out our what happened. We get the falling Mel. Let&#x27;s take are integral to that&#x27;s 1/2 You could&#x27;ve are 1/2 plus one. So that&#x27;s 3/2 and then over to do it or two crossing. So we get, um, 1/3 and then you to the power of three over to see, So that gives me 1/3. You is X squared plus 12 x to the power of 3/2 and then plus C. Okay, so here&#x27;s our solution

Ernest Castorena · Answer

this time, ask that they be integral of the rational function. 12 X men, etc. All Greg&#x27;s to the Worth minus Teoh Squirt Bus one dear. Well, partial fractions tells us that it&#x27;s from England. You have to repeated roots and X minus one X plus one. Something truly right. Your form properly, but we&#x27;ll end up with negative to over. Plus one minus five over X plus ones. Word plus two over X minus one plus one over X minus one was quick. Okay, so this is actually just the raw. That&#x27;s with positive terms for X minus one square. Honest to over. Expose one minus five. Overexpose one&#x27;s quite yet. I don&#x27;t fancy the ribs of those are pretty sure bar. We&#x27;re gonna have a natural. We have civilian. Oh, that&#x27;s my ones over X plus one squared. Like that. Time listing on this one. Yeah. And now, let&#x27;s see. So the minus one over X minus one plus fine over X plus one was an arbitrary constants E

Amy Jiang · Answer

Okay, so we want to use use up here. So why don&#x27;t we let are you equal are, um, larger polynomial, that is, Let&#x27;s let it equal X squared plus two x. But it&#x27;s what you know, if we take the derivative of that, you get to acts plus two and Indy X, So notice that we have two x puts to here and a DX so that&#x27;s equal to do you. So now let&#x27;s replace what we have. So we have do you over u to the power of four. Okay. So look to your right that as a negative exported so that you to the part of negative four. Do you? And I want to take her in a girl So he gets, um You took out negative three over negative three. Let&#x27;s see. And now, um, what&#x27;s your place? Are you with our X? So we have X squared plus two minus four to the part of negative three over the next three. Your name, plus C

Sid Wan · Answer

number eight We won&#x27;t fight ll active. Integral to CSC square eight ACS Tio with U equals a thie course. Aid the ax. So this guy equals an active into grow too. C s u square t you doing about eight? Two or eight? He was ninety one. Forcing Divorcee sees where you do You know this one by the form in there we got collective coat hand. You see that? The one force more by there. So cool one course code hands eight acs use of still back kowtow Look, cause I

Tom Greenwood · Answer

trying to do this integral. We, uh, first half Teoh dx on their the ah factor the denominator. So first we can factor out in X, and that makes that X squared minus four, which will then factor down because that&#x27;s the difference of two squares. So our decomposition to simplify this big fraction into a bunch of smaller, simpler fractions. The X cubed minus four x Well, we already know X is a factor, So there will be something over X and then the X squared minus four factors into an X minus two with a value up in the numerator and an X Plus two with Don&#x27;t value up in the numerator. So this big fraction can be separated into the three smaller fractions over X be over X minus two and see over X plus two all combined together. Question is where a B and C so we multiply by the common denominator which will get rid of all our fractions. So we get the numerator X squared plus 12 X minus four equals multiplying. Each of these the denominators going to reduce and you&#x27;re just left with the numerator times. The other two parts, so eight times The X minus two and X plus two is eight times the X squared minus four. So that&#x27;s a X squared minus four A. B times The X and the X Plus two will give us be X squared plus to be Times X and then C times the X and the X minus two is going to give us plus c X squared minus two. See X. Now we can equate our terms and the coefficients. The X squared has to equal our X squared. So one is gonna equal Hey plus B posse and then 12 X is going to equal our ex terms over here of the to be minus two C and our constant of the minus four has to equal are constant over here. Which means that we have negative four equals negative for hay, which tells us that a has two equal one. And then we can substitute that back in here and figure out being see and we end up with be equal street and C equals negative three. So this fraction and it&#x27;s interval can be rewritten as one over x plus three over X minus two minus three over x plus two DX. And now we can integrate all three of those individually pulling the constant out in front of the integral, using the one over you Do you rule? So we end up with the natural log of the absolute value of X plus three times the natural log of the absolute value of X minus two minus three times the natural look of the absolute value of X plus two. And then we will have our policy, the arbitrary constant. Yeah, and now it can use our walls of algorithms to combine all this into one algorithmic expression algorithm of the apse with value of X Times X minus two to the third power, the adding, multiplying those together and the constant of three becoming the exponents and the subtracting becomes division so divided by the X Plus two raised to the third power. Same thing, the constant becomes the exponents and we&#x27;ll under absolute value and plus our

Problem

Integrate each of the given functions. $$\int \fr…

Problem 5 Easy Difficulty

Integrate each of the given functions.
$$\int \frac{12 d x}{64+x^{2}}$$

Answer

$\frac{3}{2} \tan ^{-1}\left(\frac{x}{8}\right)+C$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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$\frac{3}{2} \tan ^{-1}\left(\frac{x}{8}\right)+C$

Basic Technical Mathematics with Calculus

Catherine R.

Heather Z.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Catherine R.

Heather Z.

Caleb E.

Michael J.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus